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Sonderforschungsbereich/Transregio 40 – Annual Report 2016 151

Impact of an Absorber Ring on AcousticDamping in Rocket Engine Chambers

By M. Schulze AND T. SattelmayerLehrstuhl für Thermodynamik, Technische Universität München

Boltzmannstr. 15, 85748 Garching b. München

A representative combustion chamber-nozzle configuration with perforated inlet andchoked outlet is numerically investigated to study the influence of absorber ring locationand cavity length on damping of acoustic amplitudes for the first transverse mode undernon-reactive ambient temperature conditions. Linearized Euler equations are solved onthe basis of a stabilized Finite-Element discretization in order to account for the meanflow effects. The results confirm experimental findings showing a reduction of damp-ing rates for an absorber ring located closely to the inlet, while an absorber ring lo-cated closely to the nozzle greatly enhances damping. Furthermore, the reduction ofthe damping rates in case of an absorber ring located closely to the inlet due to ashielding effect of the transverse mode from the highly dissipative inlet is demonstratedfor different cavity lengths and illustrated using numerical field results. Furthermore, theappearance of an additional transverse mode is found, which occurs only in the cross-section of the absorber ring. Finally, it is shown that an absorber ring located at thenozzle effectively enhances damping with a maximum for a cavity length close to thetheoretically optimum.

1. IntroductionTo guarantee tolerable thermomechanical loads on the combustion chamber walls

and the face plate, the intensification of damping of high-frequent combustion dynamicsis often pursued for reasons of low-cost and simple implementation. While most of thecomponents of the thrust system are highly critical for optimal thrust, combustion effi-ciency and weight, the margins for design modifications in favor of increasing dampingcapabilities are usually small. However, at the chamber wall, although heavily loadedthermomechanically, resonating cavities are placed in order to attenuate acoustic pres-sure amplitudes. Such absorber devices can be designed as λ/4 absorbers [1].

The influence of individual absorbers on the acoustic propagation is very complex.Inside the cavities viscous dissipation at the walls lead to an attenuation of acousticenergy. Furthermore, at the cavity opening the interaction of acoustic waves with theemanating shear layer results in the formation of vortical structures, which are convectedwith the mean flow. While the generation of vortical structures is accompanied with thereduction of acoustic energy, their impingement on the trailing edge of the cavity inletmight lead to backscattering effects and hence to an acoustic energy input.

In a recent study, the authors of [2] experimentally investigated the impact of positionand cavity length of a λ/4 absorber ring on the damping of acoustic waves. The analyseswere conducted under non-reactive ambient temperature conditions for a representative

152 M. Schulze & T. Sattelmayer

perforated

plate

siren

(optional)

hoked

boundary

absorber ring

dynami pressure

sensor

omputational

domain

air supply

DtDc

lc

FIGURE 1. Experimental test configuration and computational domain (red).

combustion chamber-nozzle configuration with a highly dissipative inlet and choked noz-zle flow outlet. It was shown that an absorber ring located closely to the inlet reducesthe damping rates for transverse modes (T1). This unexpected result was explained bya shielding effect, which decouples the acoustic mode spatially from the inlet. Conse-quently, the weaker impact of dissipation at the inlet reduces the acoustic damping. Aring located at the nozzle, in turn, enhanced the acoustic damping substantially. On thebasis of the acoustic power balance at the absorber ring element, it was shown that theabsorber ring itself is not capable of amplification, which emphasizes the shielding ef-fect hypothesis. In rocket engines with reactive flows absorber rings are conventionallylocated in the proximity of the injection area in order to attenuate acoustic amplitudes inthe source region of the coupling between acoustic propagation and combustion. Con-sequently two counteracting effects have to be studied and compared to find the optimaldesign of damping cavities.

The study at hand aims for a detailed investigation of the impact of absorber ringlocation and cavity length on the damping of acoustic waves including the shieldingeffect hypothesis by using numerical procedures. Essentially, the same configurationas presented in [2] is used. The axial dimension of the chamber is, however, reducedin order to gain a more realistic length to diameter ratio. At the inlet, the perforatedplate provides strong damping of acoustic waves due to the aeroacoustic interactionof the transverse modes with emanating jet structures [3]. Two configurations with theabsorber ring located at the inlet and at the nozzle are considered.

The investigation is presented as follows. First, the test configurations are describedand geometrical as well as operational details are given. Afterwards the computationalprocedure for the determination of the frequency response as well as the damping ratesof the first transverse mode is explained. The obtained results are presented and dis-cussed in the subsequent chapter. Finally, the main conclusions are given and futurework possibilities are motivated.

Impact of an Absorber Ring on Acoustic Damping in Rocket Engine Chambers 153

TABLE 1. Summary of geometrical and operational conditions.Symbol Value

Chamber diameter Dc 92 mm

Throat diameter Dt 60 mm

Chamber length lc 107 mm

Distan e perforated plate - absorber ring for position 1 lP1

36 mm

Distan e perforated plate - absorber ring for position 2 lP2

102 mm

Mass �ow of pressurized air m 1.15 kg/s

Chamber Ma h number Mac 0.25

Chamber pressure (stati ) pc 1.65 bar

Cavity length lA 0-45 mm

Theoreti al optimal avity length lA,optimal

39.24 mm

Absorber diameter DA 8 mm

Perforated plate, hole diameter 3 mm

Perforated plate, separation 4 mm

Perforated plate, thi kness 2 mm

Perforated plate, open area ratio 51 %

Perforated plate, hole Ma h number 0.4

2. Test ConfigurationThe experimental test configuration is shown in Fig. 1. The chamber comprises a

cylindrical part with a diameter of Dc = 92 mm and length of 107 mm as well as aconvergent-divergent nozzle with a throat diameter of Dt = 60 mm. At the chamber inleta perforated plate with an open area ratio of 51% is mounted to generate a homoge-neous flow of air. At the same time the perforated plate imitates the face plate of realisticrocket engine configurations with a high number of injectors. In the single holes a Machnumber of 0.4 is reached. Further details of the perforated plate can be found in Tab.1. It has been shown in previous studies that damping of the T1 mode is tremendouslyincreased through the interaction of the acoustic motion with the shear layers behind theperforated plate [4].

A mass flow of m = 1.15 kg/s air at ambient temperature is used leading to a Machnumber of 0.25 in the chamber and choked conditions establish in the nozzle. The staticpressure is given by pc = 1.65 bar. The supersonic flow in the divergent part of thenozzle provides a natural decoupling of the chamber acoustic from the environment.Upstream of the perforated plate an annular slit establishes a choked boundary to de-couple also the air supply system. Therefore, experimentally determined damping ratesfor the T1 mode contain only dampening effects occurring within the chamber and noz-zle.

To study the influence of absorber cavities on the acoustic damping of the T1 mode,the chamber is equipped with an absorber ring which comprises 20 λ/4-cavities witha diameter of DA = 8 mm. The cavities are equidistantly distributed in circumferentialdirection and set to the same length, which is varied between 30 mm and 45 mm insteps of 5 mm. At first, to simulate a typical rocket engine arrangement, the absorberring is located closely to the face plate, i.e. to the perforated plate. This absorber ringlocation is referred to as position 1 (P1), see Fig. 2 (left). The distance between theperforated plate and the absorber ring amounts lP1 = 36 mm. Experimental validationdata for this arrangement in terms of frequency response as well as eigenfrequenciesand damping rates for the T1 mode is taken from [5].

Experimental findings showed that an absorber ring arranged according to configura-


absorber ring

lP1

lP2

lc

lA

FIGURE 2. Experimental test configuration and computational domain (red) for absorber ringpositions P1 (left) and P2 (right).

tion P1 reduces the damping rate of the T1 mode [5]. Further experimental evaluationsindicate significant influence of the absorber ring’s axial location on the damping prop-erties [2,5]. To investigate this phenomena, a second configuration P2 is considered aswell, where the absorber ring is located at position 2, which is close to the nozzle, seeFig. 2 (right). In this case lP2 is 102 mm.

3. Computational Procedure3.1. Frequency Space Transformed Linearized Euler Equations

The numerical procedure is based on a CFD/CAA hybrid approach, where the primaryvariables q are decomposed into a statistically mean state () and a fluctuating part ()′ interms of space xi and time t, viz.

q(xi, t) = q(xi) + q′(xi, t). (3.1)

Assuming the fluctuations to be small compared to the mean state, the linearization ofthe governing equations for the fluctuating part is justified. If diffusion of momentum andheat is negligible, as it is usually the case for acoustics, linearized Euler equations aresufficient. In the present case, the influence of entropy can also be neglected, leadingto a further reduced set of equations by imposing the isentropic condition, viz.

p′ = c2ρ′, (3.2)

where p denotes pressure, ρ density and c speed of sound. On the basis of linear acous-tic propagation, the fluctuating quantities can be described in terms of space and timedependency separately

q′(xi, t) = q(xi)eiωt, (3.3)

where ω denotes the angular frequency. Through the decomposition of the fluctuatingpart into time and space dependence the isentropic linearized Euler equations in fre-


quency space are finally given by

iωρ+ ρ∂ui∂xi

+ ui∂ρ

∂xi+ ρ

∂ui∂xi

+ ui∂ρ

∂xi= 0, (3.4)

iωui + uj∂ui∂xi

+ uj∂ui∂xj

+c2

ρ

∂ρ

∂xi+

ρ

ρ2∂p

xi= 0, (3.5)

where ui states the i-th velocity component. In this study, the mean flow quantities aredetermined by means of a standard RANS simulation using ANSYS CFX 14.5 and thefluctuating quantities are determined from acoustic simulations in a subsequent step.

In the present work a Finite Element discretization is used. It is known that the stan-dard Galerkin Finite Element scheme for convective dominated flow problems exhibitsunphysical oscillations in the solution, see e.g. [6]. The authors of [7] show that stabil-ity can be achieved while solving the consistent set of equations when using stabilizedFinite Element methods. An overview is given in [6]. Here the so-called Galerkin/least-squares (GLS) technique is applied.

3.2. Computational Setup and Absorber Model

The computational domain is shown in Fig. 2 in red and a detailed overview of the ap-plied boundary conditions as well as of the grid topology is illustrated in Fig. 3. Previousstudies have shown that eigenfrequencies and damping rates of the T1 mode in theconfiguration without absorber ring can be predicted reasonably accurate on the basisof a reflection coefficient for the entire air supply system both in time domain [8] andfrequency space [9]. For this reason the volume upstream of the perforated plate isnot included in the simulations. Instead, the reflection coefficient Rpp is imposed at thecorresponding inlet surface as boundary condition.

Acoustic excitation is necessary to determine the frequency response. In the experi-mental procedure, broadband excitation using natural flow noise is employed to providea fast procedure for the determination of the Power Spectral Density (PSD) [2, 5]. Thehigh Mach number flow, especially in the single holes of the perforated plate, generatesa satisfying acoustic signal-to-noise ratio. In the numerical evaluations, however, flownoise is not included. Acoustic excitation is provided by velocity modulation on a smallcircular region, which is eccentrically located at the inlet surface, see red circle at theinlet in Fig. 3. The location for excitation at the inlet is advantageous, since it providesa larger distance to the absorber ring and reduces unwanted interactions with the cav-ities. Furthermore, it models the flow noise excitation more realistically. The boundarycondition reads

uini|exc. inlet = uv. (3.6)

Modeling the velocity excitation amplitude uv is difficult, since the frequency spectrumof the flow noise modulation is unknown. The noise generation, however, is primarilybased on turbulence, which is dominated by the dynamics of the emanating shear layerat the single holes of the perforated plate. According to [10], the shear layer behaviorcan be characterized by the Strouhal number St. It is shown that for St < 0.25 the shearlayer is unstable and strong excitation can be expected. In turn, stable behavior is foundfor St > 0.25. According to [11], the Strouhal number can be approximated by

St =ωdBL

cM, (3.7)


wall

uini = 0

ane hoi

R = 0

absorber ring

RAR

perforated plate

Rpp

velo ity ex itation

u = u∗

FIGURE 3. Numerical grid and boundary conditions for the acoustic simulation of thenon-reactive test case with absorber ring.

where the boundary layer thickness dBL at the separation edge is defined as 10 % of theupstream pipe radius in case of an area expansion. For the perforated plate the singlehole radius is used. For the highest considered frequency of 3000 Hz, a Strouhal numberof 0.021 is found, which is well below the critical St number. In consequence, a constantamplitude modulation for acoustics is applied. The amplitude of the velocity excitation isscaled such that the resulting maximum amplitude in the frequency response coincideswith the maximal amplitude in the experimental spectrum. For linear acoustics, such ascaling procedure is allowed.

For the determination of the frequency response, the reflection coefficient of the per-forated plate needs to be applied for the considered entire frequency range. Below thecut-on frequency of the T1 mode, longitudinal modes dominate the acoustic field. Fur-thermore, it is assumed that the transverse mode motion dominates in the high fre-quency regime and the reflection coefficient for the T1 mode can therefore directly beapplied. This assumption is justified by the fact that the nozzle provides high damp-ing for longitudinal modes. The experimentally determined reflection coefficient is takenfrom [5].

The detailed absorber inlet regions are not resolved but the reflection coefficient ofeach single absorber RSA is mapped onto the corresponding shell surface, see Fig. 4.The mapping is represented in form of an equivalent reflection coefficient for the ab-sorber ring RAR. The shell width is set equal to the cavity diameter.

The transformation of a reflection coefficient of a single absorber RSA into RAR isbased on the conservation of the fluctuating mass and reads

RAR =ASZSA −NSAASA

ASZSA +NSAASA, (3.8)

where ZSA denotes the impedance of a single absorber. Furthermore, AS and ASA de-note the areas of the shell surface and single absorber inlet and NSA is the number ofabsorbers in the ring.

The reflection coefficient of the absorber cavities is not given experimentally and istherefore determined from theoretical models. Different absorber models are proposede.g. in [12–19]. All these models contain certain parameters which can either be linkedto further relations or need to be calibrated on the basis of experimental results or so-phisticated numerical procedures. The presented results are based on the quarter wave


RSA → RAR

FIGURE 4. Mapping of the single absorbers onto the circular shell surface.

model suggested in [19]. It defines the impedance by

ZSA ρ c = 2

(1 + ǫnl +

lADA

)√2µρω + iρ c cot

(le ω

c

), (3.9)

where µ denotes viscosity. In Eq. (3.9), ǫnl denotes a non-linear resistance factor and lean effective length, which takes inertia effects into account and is approximately givenby [20]

le =4

3πDA ≈ 0.424DA. (3.10)

The non-linear resistance factor is a major source of uncertainty. As originally pro-posed in [21] and discussed in [12], ǫnl may vary over several orders of magnitude de-pending on the Sound Pressure Level (SPL). In the present test case, maximal pressureamplitudes of up to approximately 167 dB (5000 Pa) are observed in the chamber closeto the eigenfrequencies, which results in ǫnl = O(102) [4]. For other frequencies, theSPL is considerably lower and ǫnl reduces accordingly. Throughout this investigation thenon-linear factor is set to ǫnl = 10.

The reflection coefficient RAR for the considered cavity lengths is shown in Fig. 5in terms of amplitude and phase. At the quarter wave eigenfrequency of the cavities,which is given under the conditions that the imaginary part of the impedance in Eq. (3.9)is zero, low reflection amplitudes and an anti-phase behavior of ingoing and outgoingwaves are observed. Hence, for T1 eigenfrequencies of the system close to the quarterwave eigenfrequency of the cavities, acoustic dissipation can be expected. However, theinfluence of the absorber ring is not only restricted to the quarter wave eigenfrequency,but a rather broad spectrum of low amplitude and phase modulation might also influenceother modes.

It is known that the absorber damping properties are based on different physical pro-cesses [12]. Within the cavities, viscous boundary layer dissipation reduces acousticamplitudes. In case of a standing mode pattern in the cavities for resonant frequen-cies, viscous dissipation is especially effective. At the cavity mouth, acoustic fluctua-


ϕ(R

AR)

|RAR|

L30

L35

L40

L45

0.6

0.8

1.0

ππ2

0

-

π2

-π1000 2000 3000 4000 1000 2000 3000 4000

frequen y [Hz℄ frequen y [Hz℄

FIGURE 5. Amplitude (top) and phase (bottom) of the absorber ring reflection coefficient.

tions modulate shear layers instabilities, which produce vortical structures and therebydissipate acoustic energy. Furthermore, at resonant conditions inside the cavities, anout-of-phase relation between ingoing and outgoing waves is established, see Fig. (5).Consequently, a destructive effect of the outgoing wave on the transverse mode is real-ized, which directly reduces the T1 amplitude in the chamber.

The dissipation of acoustic energy due to both boundary layer effects and the gen-eration of vortical structures is modeled in form of viscous dissipation in Eq. (3.9). Thedifferent coefficients including the non-linear resistance can therefore be seen as en-hanced viscous dissipation factors.

With an increase of the cavity length, the quarter wave eigenfrequency is reducedand the range of minimum reflection amplitude is shifted towards lower frequencies.Optimized damping can therefore be expected if the T1 system eigenfrequency and thequarter wave eigenfrequency coincide. The cavity length is then given by

lA =c

4fT1

=π

4s10Dc ≈ 0.427Dc, (3.11)

which results in a theoretical optimum length for the considered chamber configurationof lA,opt = 39.24 mm.

Studies from literature [2, 22, 23] show that the decoupled consideration of chamberand absorber dynamics is not the optimal design procedure. Essentially, the coupledsystem determines the T1 eigenfrequency and absorbers should be tuned such thatthese eigenfrequencies are damped in an optimal way. The development of powerfulabsorber rings can therefore only be performed iteratively by reevaluating the systemeigenfrequencies and damping rates after each modification of the absorber ring de-sign and Eq. (3.11) can only be used as an initial approximation. For this purpose, fastprocedures are desirable.

The computational grid used in this study is shown in Fig. 3. The resolution is refinedin the vicinity of the excitation and close to the absorber ring in order to provide smoothtransition to the neighboring wall conditions.

4. Results and Discussions4.1. Frequency Response for Absorber Ring Position 1

The frequency response is studied for the absorber ring position P1, cf. Fig. 2. For thisarrangement, the two absorber lengths lA = 35 mm and lA = 40 mm are considered.Furthermore, the experimentally measured frequency response without absorbers (i.e.lA = 0 mm) is also taken into account as reference configuration in order to illustrate theeffects of the absorber ring. The frequency responses for the different absorber lengthsare shown in Fig. 6.


exp.

num.

exp. ref.

1000 2000 3000 1000 2000 3000

0

50

100

35 mm 40 mm

frequen y [Hz℄ frequen y [Hz℄

T1−

T1+d

T1

|p|

[

P

a

℄

FIGURE 6. Frequency response of |p| [Pa] for 35 mm (left) and 40 mm cavity length (right) incomparison to the reference case without absorber ring: experimental findings and numericalresults.

First, the frequency response with absorber ring is compared to the numerical findings(num.) and the experimental results (exp.). Overall, good agreement is obtained. Char-acteristic peak values in the spectrum and corresponding eigenfrequencies are verywell predicted, which implies that excitation amplitude and system damping character-istics are adequately modeled when the amplitudes are dominated by an eigenmode.For non-resonant frequencies, however, discrepancies are observed. Apparently, theassumption of negligible contributions from longitudinal modes is not fully correct withinthese frequency ranges.

In the next step, the frequency responses with absorber ring are compared to thefindings of the reference case without absorber ring. It can be seen in Fig. 6, that withabsorber ring there are two peaks near to the T1 frequency, on the low and high fre-quency sides respectively. The separated modes are commonly referred to as T1

− andT1

+ [2,5,22,24]. The nomenclature T1− denotes an acoustic mode, which is not present

for the system without absorber ring. For the sake of clarity, the notation T1+ is also in-

troduced.To the author’s knowledge, the T1

− and T1+ mode have not yet been investigated

with convection involved [22]. The LEE results presented in this paper allow a moredetailed analysis. For that purpose, the pressure shapes for different modes identifiedin the spectra are studied next, see Fig. 7.

Clearly, the absorber ring significantly influences the mode shapes. While an approx-imately homogeneous pressure distribution in the chamber is established for the T1

mode in the reference case (Fig. 7 left), high pressure amplitudes are only found di-rectly at the absorber ring surface for the T1

− mode (Fig. 7 middle). Therefore, the T1−

mode exists primarily in the absorber ring section. The increased cross-sectional areadue to the absorber cavities allows for the appearance of the T1

− mode at reducedfrequencies, which is in accordance with the frequency response data. With increasingcavity lengths, the T1

− mode frequency decreases, since the cross-sectional area be-comes larger. In fact, for an absorber length of lA = 35 mm, the eigenfrequency of theT1

− mode is approximatively 1880 Hz and reduces in case of an absorber length of lA =40 mm to approximately 1760 Hz.

Fig. 7 (right) reveals a mode shape, which exists predominantly downstream of theabsorber ring and is commonly identified as T1

+. However, to be more precise, thismode is referred to as T1

+d for future reference indicating that another mode exists,

which primarily appears upstream of the absorber ring. In terms of eigenfrequency, ashift of the T1

+d mode towards higher frequencies is observed in the spectra, see Fig. 6.

This behavior can be explained by the reduced axial extent of the mode shape due tothe influence of the absorber ring.


T1 T1−

T1+d

0 1

FIGURE 7. Mode shapes for configuration P1 in terms of normalized |p| under excitation for thepeak frequencies in the spectra, cf. Fig. 6. (left): Reference case without absorber. (middle) and(right): Modes shapes with an absorber cavity length of 35 mm.

2500

2000

1500

800

400

0

0 10 20 30 40 50 0 10 20 30 40 50

exp.

num. part. re�e t.

num. neutral

f T

1+ d

[

H

z

℄

λT

1+ d

[

r

a

d

/

s

℄

lA

[mm℄ lA

[mm℄

FIGURE 8. Eigenfrequency (left) and damping rate (right) of the T1+d mode for configuration P1:

Experimental findings and numerical results both for the partially reflecting inlet and the neutralinlet.

4.2. Eigenfrequencies and Damping Rates for Absorber Ring Position 1

The capabilities of the LEE to determine the eigenfrequency and damping rates underthe influence of the absorber ring for different cavity lengths is studied next. In this study,the absorber ring is located close to the perforated plate, i.e. in position 1, cf. Fig. 2.

The authors of [2,5] reported high damping rates for the T1− mode for all considered

cavity lengths. Probably, the strong attenuation of the T1− mode is attributed to the

fact that its eigenfrequency is cut-off in the regions next to the absorber ring sectionleading to a high imaginary part of the associated wave number. Consequently, theT1

− mode is uncritical in terms of thermoacoustic instabilities and therefore not furtherconsidered. A more detailed analysis can be found in [2]. In this study, only the T1

+ istaken into account. First, the focus is on the T1

+d mode, which shows high amplitudes

predominantly in the section downstream of the absorber ring, cf. Fig. 7.In the acoustic simulation, the excitation boundary condition is replaced by the reflec-

tion coefficient of the perforated plate, cf. Fig. 3, and the eigenvalue problem is solvedusing different cavity lengths. The reflection coefficients of the perforated plate and ofthe absorber ring are evaluated at the frequency fT1

+d

.

The experimental findings and numerical results for the T1+d mode in terms of eigen-

frequencies and damping rates for increasing cavity lengths are presented in Fig. 8. Alto-gether, very good agreement can be observed between the experimental data (exp., redline) and the numerically determined eigenfrequencies and damping rates (num. part.reflect., blue line). The good comparison again shows that the LEE together with theapplied boundary treatment is well suited for the predictions of absorbers under grazingflow conditions for transverse acoustic motion. It can also be concluded that the evalua-tion procedure of the reflection coefficients of the perforated plate and the absorber ringat fT1

+d

is adequate.The numerically determined eigenfrequencies show constant but small discrepancies

to the experimental data, which are, however, below 2 % deviation. The small discrepan-


cies originate probably from the missing influence of dynamics of the air supply systemfurther upstream. For 35 mm excellent agreement is observed. Concerning the dampingrates, excellent agreement is observed for 35 mm and 40 mm, while small deviationsare found for 30 mm and 45 mm. These discrepancies can also be explained by themissing influence of the dynamics of the air supply system and accompanied changesin eigenfrequency and therefore in RAR elevated at ωT1

+d

.

The results show, that the T1+d eigenfrequency is higher than the eigenfrequency of

the T1 mode, which occurs without absorber ring (i.e. lA = 0 mm). The maximum differ-ence of is 9 %. For increasing cavity length the shift decreases and the eigenfrequencytends towards the reference T1 mode frequency.

In contrary to expectations, the damping rates of the T1+d mode are significantly re-

duced for the considered range of cavity lengths in comparison to the reference casewithout absorber ring. The strongest reduction of 44 % is found for 35 mm. For highercavity lengths the damping rate tends towards the reference value of the T1 mode again.

Reduced damping rates could be explained by amplification properties of the absorberring. As a result from the interaction of grazing flow and acoustic fluctuations at the cavitymouth, so-called whistling effects might in principle lead to an energy transfer into theT1

+d mode [25]. On the basis of acoustic intensity balancing, [5] and [2] show, however,

that the considered absorber ring configuration is not capable of acoustic amplification.Next, the T1

+d mode in terms of pressure amplitude |p| is studied in Fig. 9. The mode

exists predominantly downstream of the absorber ring and shows only small amplitudesin the upstream part. The spatial separation effect is especially intense for 35 mm. Forthis cavity length, the strongest reduction of damping rates is found, cf. Fig. 8. For 45mm length, in turn, higher pressure amplitudes are present in the upstream part and aweaker reduction of the damping rate is observed.

The spatial separation of the T1+ mode can be explained by the impact of the ab-

sorbers on the mode propagation in the absorber ring cross-section. Two mechanismsare identified: On the one hand, the T1

+d is not in resonance within the absorber ring

section due to enlarged virtual diameter in this region. On the other hand, high trans-verse velocities at the absorber inlets and accompanied dampening effects restrict theaxial propagation across the ring element. The velocities are illustrated in Fig. 9 (right)in form of arrows at the absorber ring surface. Furthermore, the impact on R(p) showsthe distinct separation of the mode structure at the absorber interface.

Due to the spatial separation, the axial dimension of the T1+d mode is reduced. A

shortened longitudinal component leads to higher eigenfrequencies, which is confirmedin Fig. 8 (left). For increasing cavity lengths, the separation effects gets weaker and theeigenfrequencies therefore converge towards the reference T1 mode frequency.

Furthermore, the spatial separation effect shields the T1+d mode from the perforated

plate. As the main contribution to acoustic dissipation originates from the inlet, its damp-ing effect is reduced accordingly. Therefore, the reduction of the damping rate resultsfrom the weakened impact of the perforated plate. The stronger the absorber ring’s influ-ence, the more intense the separation and consequently the reduction of the dampingrates. This explains the minimum value found for a cavity length of 35 mm.

In order to support the hypothesis of the spatial mode separation, the absorber ring’simpact is investigated for an energetically neutral inlet condition for configuration P1. Forthis purpose, a zero varying mass flux condition is prescribed at the inlet according to

ˆm = ρu+ ρu = 0. (4.1)


30 mm 35 mm

40 mm 45 mm

35 mm; R(p) andvelo ity ve tors on the

absorber ring surfa e

-1 1

0 1

FIGURE 9. Mode shapes T1+d for different cavity lengths lA in terms of normalized |p| and R(p)

for 35 mm length (right) for configuration P1.

30 mm 35 mm

40 mm 45 mm

T

+1u

0 1

FIGURE 10. Mode shapes for different cavity lengths lA and an neutral inlet boundary conditionin case of configuration P1: T1

+d and T1

+u (right) modes in terms of normalized |p|.

In this configuration, the shielding effect may not result in reduced damping rates, sincethe inlet does not provide acoustic dissipation. The corresponding mode shapes areillustrated in Fig. 10. The numerical results in terms of eigenfrequencies and dampingrates are presented in Fig. 8 (num. neutral, green dashed line) together with the findingsfor the partially reflecting inlet data.

Neutral inlet conditions do not considerably influence the eigenfrequencies. Overall,good agreement between the experimental results and the numerical findings on thebasis of the partially reflecting boundary condition as well as on the neutral inlet isobtained.

In comparison to the reference case with neutral inlet and a damping rate of λT1=

37 rad/s, the incorporation of the absorber ring leads to tremendously increased damp-ing rates for all cavity lengths. Consistently, the strongest impact is provided by a cavitylength of 35 mm. In this situation, the damping rate increases by a factor of approx-imately 10. The damping rates even exceed the values found in case of a partiallyreflecting inlet. Also, for increasing cavity lengths the damping rate tends towards thereference again.


30 mm 35 mm

40 mm 45 mm

35 mm; R(p) andvelo ity ve tors on the

absorber ring surfa e

0 1

FIGURE 11. Mode shapes T1+u for different cavity lengths lA and partially reflecting inlet in terms

of normalized |p| and R(p) for 35 mm length (right) for configuration P2.

Interestingly, as the inlet does not provide damping, higher amplitudes are observedin the upstream part compared to the case with partially reflecting inlet, cf. Fig. 9 withFig. 10. Furthermore, in the upstream part another mode is found, which is consequentlyreferred to as T1

+u . Since this mode is subjected to the highly dissipating inlet in reality

as well as to the absorber ring, damping rates are significantly higher than for the T1+d

mode. Therefore, the T1+u mode is not considered in more detail for configuration P1.

In summary, an absorber ring increases the damping rates for neutral inlet conditions,while the combination with a highly dissipating inlet causes reduced damping rates com-pared to the respective reference configuration. In both cases, the shielding effect de-couples the T1

+d mode from the inlet with very different influences on the damping rates.

Furthermore, the shielding effect hypothesis is supported by the fact that approximatelyequivalent damping rates are found for 30 mm and 35 mm cavity length, for which thespatial separation is most intense, cf. Fig. 8 (right). In this situation the T1

+d mode is

almost completely separated and becomes therefore independent from the inlet.Overall, the highest damping rates are found for a shorter cavity length than theo-

retically predicted according to Eq. 3.11, which is in accordance with results presentedin [23] and [2].

4.3. Eigenfrequencies and Damping Rates for Absorber Ring Position 2

As explained before, the upstream T1+u mode is subject to significant damping for config-

uration P1 with the partially reflecting inlet condition. Locating the absorber ring furtherdownstream, however, increases the mode volume and therefore influences the modeseparation. In this section, the absorber ring is located at the end of the chamber closeto the nozzle and the impact on eigenfrequencies and damping rate is investigated onthe basis of the partially reflecting inlet condition. For configuration P2, experimentalreference values for a validation are not yet available. Therefore, only numerical findingsare presented and discussed.

In Fig. 11 the T1+u mode shapes are presented for different cavity lengths. The mode

shapes exist in almost the entire chamber and are separated from the nozzle. High-est amplitudes are present directly at the inlet. For increasing cavity lengths, significantpressure amplitudes reach further into the chamber, indicating that the strongest sepa-

164 M. Schulze & T. Sattelmayer2500

2000

1500

800

400

0

0 10 20 30 40 50 0 10 20 30 40 50

f T

1+ u

[

H

z

℄

λ

T

1+ u

[

r

a

d

/

s

℄

lA

[mm℄ lA

[mm℄

FIGURE 12. Numerical results: Eigenfrequency (left) and damping rate (right) of the T1+u mode

for configuration P2.

ration exists for 30 mm. For 45 mm, the lowest intensity of separation is observed dueto the weak influence of the absorber ring.

The T1+d mode exists in the nozzle part (not shown). However, due to its small axial

extent and the varying cross-sectional area in the nozzle, the T1+d mode is strongly

damped and therefore not considered here.The influence of the absorber ring on the eigenfrequency of the T1

+u mode is pre-

sented in Fig. 12 (left). In comparison to configuration P1, the shift in eigenfrequency toincreased values is larger, which is explained by a stronger reduction of axial extent dueto the decoupling of the entire nozzle part. Furthermore, a trend towards the referencesolution is not present. The decoupling of the nozzle part is quite significant resulting tostrong frequency shifts for already low influences of the absorber ring.

In Fig. 12 (right) a significant increase of the damping rate of the T1+u mode is visible.

Especially for 35 mm and 40 mm extremely high values are found. For even higher cavitylengths, the damping rate tends towards the reference value. Similar observations werereported in [2].

Since in configuration P2 the shielding of the T1+u mode from the dissipating inlet

is not present, both inlet and absorber ring contribute to damping. The damping ratesare thus significantly higher than for configuration P1. In consequence, in case of ahighly dissipating perforated plate, enhanced damping due to absorber devices can beachieved by realizing a substantial distance between absorber ring and the inlet.

The mode distribution in the cross-section of the absorber ring position for a cavitylength of 35 mm is shown in Fig. 11 (right). The transverse velocity arrows indicate astrong interaction with the cavities. Furthermore, R(p) shows that directly at the cav-ity mouth pressure is low for this phase instant, which again illustrates the decouplingmechanism between upstream and downstream modes.

5. Conclusions and Future Work PossibilitiesIn this work the impact of an absorber ring in terms of its axial location and cavity

length was studied on the basis of numerical results. For their generation a stabilizedFinite-Element discretization of the linearized Euler equations are solved in order to ac-count for mean flow effects. Frequency response and damping rates for the first trans-verse mode are found in good agreement with experimental reference data. Significantchanges in the mode shape should be expected due to the influence of absorber rings.Spatial mode separation and accompanied shielding effects tremendously change thesystem dynamics possibly leading to increased eigenfrequencies and reduced dampingrates when highly dissipating boundary conditions are decoupled from the mode.

In terms of combustion instabilities, the influence of absorber rings on the damping


but also on the dynamic flame response should be investigated next. It needs to be eval-uated whether the apparent shielding effect of dissipating boundaries might in fact leadto less stable systems. However, due to the accompanied reduction of acoustic ampli-tudes near the face plate, absorber rings might also significantly reduce the dynamicflame response, which can in turn increase stability margins.

AcknowledgmentsFinancial support has been provided by the German Research Foundation (Deutsche

Forschungsgemeinschaft – DFG) in the framework of the SonderforschungsbereichTransregio 40.

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